Beyond the Sottile-Sturmfels degeneration of a semi-infinite Grassmannian

TL;DR

This paper explores new toric degenerations of semi-infinite Grassmannians, providing a novel interpretation of existing degenerations and introducing a new construction based on poset polytopes, along with a basis for their coordinate rings.

Contribution

It offers a new interpretation of the Sottile-Sturmfels degeneration and constructs a novel toric degeneration for semi-infinite Grassmannians using poset polytopes.

Findings

Reinterpreted Sottile-Sturmfels degeneration as a toric variety of an order polytope.

Constructed a new toric degeneration in the semi-infinite case.

Introduced semi-infinite PBW-semistandard tableaux as a basis.

Abstract

We study toric degenerations of semi-infinite Grassmannians (a.k.a. quantum Grassmannians). While the toric degenerations of the classical Grassmannians are well studied, the only known example in the semi-infinite case is due to Sottile-Sturmfels. We start by providing a new interpretation of the Sottile-Sturmfels construction by finding a poset such that their degeneration is the toric variety of the order polytope of the poset. We then use our poset to construct and study a new toric degeneration in the semi-infinite case. Our construction is based on the notion of poset polytopes introduced by Fang-Fourier-Litza-Pegel. As an application we introduce semi-infinite PBW-semistandard tableaux, giving a basis in the homogeneous coordinate ring of a semi-infinite Grassmannian.